matrices and algbra

1

Linear Algebra

Systems of Linear Equation and Matrices

CHAPTER 1

FASILKOM UI 05

Linear Algebra - Chapter 1 [YR2005] 3

Introduction ~ Matrices

Information in science and mathematics is often organized into rows and columns to form rectangular arrays.

Tables of numerical data that arise from physical observations

Example: (to solve linear equations)

Solution is obtained by performing appropriate operations on this matrix

− 412

315

42

35

=−=+yx

yx

1.1 Introduction to

Systems of Linear Equations


Linear Equations

In x y variables (straight line in the xy-plane)

where a1, a2, & b are real constants,

In n variables

where a1, …, an & b are real constants

x1, …, xn = unknowns.

Example 1 Linear Equations

The equations are linear (does not involve any products or roots of variables).

byaxa =+ 21

bxaxaxa nn =+++ ...2211

73 =+ yx 1321 ++= zxy 732 4321 =+−− xxxx


Linear Equations

The equations are not linear. A solution of is a sequence

of n numbers s1, s2, ..., sn Э they satisfy the equation when x1=s1, x2=s2, ..., xn=sn (solution set).

Example 2 Finding a Solution Set

1 equation and 2 unknown, set one var as the parameter (assign any value)

or

1 equation and 3 unknown, set 2 vars as parameter

53 =+ yx 423 =+−+ xzzyx xy sin=

bxaxaxa nn =+++ ...2211

124 =− yx

212, −== tytx tytx =+= ,4

121

574 321 =+− xxx

txsxtsx ==−+= 321 ,,745


Linear Systems / System of Linear Equations Is A finite set of linear equations in the vars x1, ..., xn

s1, ..., sn is called a solution if x1=s1, ..., xn=sn is a solution of every equation in the system.

Ex.

x1=1, x2=2, x3=-1 the solution

x1=1, x2=8, x3=1 is not, satisfy only the first eq.

System that has no solution : inconsistent System that has at least one solution: consistent

Consider:

493

134

321

321

−=++−=+−

xxx

xxx

00,:

00,:

112222

111111

≠∧≠=+≠∧≠=+bacybxal

bacybxal


(x,y) lies on a line if and only if the numbers x and y satisfy the equation of the line. Solution: points of intersection l1 & l2

l1 and l2 may be parallel:

no intersection, no solution

l1 and l2 may intersect

at only one point: one solution

l1 and l2 may coincide:

infinite many points of intersection, infinitely many solutions

Linear Systems

x

x

y1l 2l

x

y

x

y

1l 2l

21 & ll


Linear Systems

In general: Every system of linear equations has either no solutions, exactly one solution, or infinitely many solutions.

An arbitrary system of m linear equations in n unknowns:a11x1 + a12x2 + ... + a1nxn = b1

a21x1 + a22x2 + ... + a2nxn = b2

am1x1 + am2x2 + ... + amnxn = bm

x1, ..., xn = unknowns, a’s and b’s are constants aij, i indicates the equation in which the coefficient

occurs and j indicates which unknown it multiplies


Augmented Matrices

Example:

Remark: when constructing, the unknowns must be written in the same order in each equation and the constants must be on the right.

mmnmm

n

n

baaa

baaa

baaa

21

222221

111211

0563

1342

92

321

321

321

=−+=−+

=++

xxx

xxx

xxx

−−

0563

1342

9211


Augmented Matrices

Basic method of solving system linear equations Step 1: multiply an equation through by a nonzero

constant. Step 2: interchange two equations. Step 3: add a multiple of one equation to another.

On the augmented matrix (elementary row operations): Step 1: multiply a row through by a nonzero

constant. Step 2: interchange two rows. Step 3: add a multiple of one equation to another.


Elementary Row Operations (Example)

r2= -2r1 + r2

r3 = -3r1 + r3

0563:

1772:

92:

3

2

1

=−+−=−=++

zyxr

zyr

zyxr

0563:

1342:

92:

3

2

1

=−+=−+

=++

zyxr

zyxr

zyxr

−−

0563

1342

9211

−−−

0563

17720

9211

27113:

1772:

92:

3

2

1

−=−−=−=++

zyr

zyr

zyxr

−−−−

271130

17720

9211


Elementary Row Operations (Example) r2 = ½ r2

r3 = -3r2 + r3

r3 = -2r3

27113:

:

92:

3

217

27

2

1

−=−−=−=++

zyr

zyr

zyxr

−−−−

271130

10

9211

217

27

23

21

3

217

27

2

1

:

:

92:

−=−−=−=++

zr

zyr

zyxr

−−−−

23

21

217

27

00

10

9211

3:

:

92:

3

217

27

2

1

=−=−=++

zr

zyr

zyxr7 172 2

1 1 2 9

0 1

0 0 1 3

− −


Elementary Row Operations (Example) r1 = r1 – r2

r1 = -11/2 r3 + r1

r2 = 7/2 r3 + r2

Solution:

3:

:

:

3

217

27

2

235

211

1

=−=−

=+

zr

zyr

zxr

−−3100

10

01

217

27

235

211

3100

2010

1001

3:

2:

1:

3

2

1

===

zr

yr

xr3:

:

1:

3

217

27

2

1

=−=−

=

zr

zyr

xr

−−3100

10

1001

217

27

1.2 Gaussian Elimination


Echelon Forms

Reduced row-echelon form, a matrix must have the following properties: If a row does not consist entirely of zeros the the

first nonzero number in the row is a 1 = leading 1 If there are any rows that consist entirely of zeros,

then they are grouped together at the bottom of the matrix.

In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.

Each column that contains a leading 1 has zeros everywhere else.


Echelon Forms

A matrix that has the first three properties is said to be in row-echelon form.

Example: Reduced row-echelon form:

Row-echelon form:

−

−00

00,

00000

00000

31000

10210

,

100

010

001

,

1100

7010

4001

1 4 3 7 1 1 0 0 1 2 6 0

0 1 6 2 , 0 1 0 , 0 0 1 1 0

0 0 1 5 0 0 0 0 0 0 0 1

− −


Elimination Methods

Step 1: Locate the leftmost non zero column

Step 2: Interchange r2 ↔ r1.

Step 3: r1 = ½ r1.

Step 4: r3 = r3 – 2r1.

−−−−−

156542

281261042

1270200

−−−−−

156542

1270200

281261042

−−−−−

156542

1270200

1463521

−−−−

29170500

1270200

1463521


Elimination Methods

Step 5 : continue do all steps above until the entire matrix is in row-echelon form.

r2 = -½ r2

r3 = r3 – 5r2

r3 = 2r3

−−−−

−

29170500

60100

1463521

27

−−

−

10000

60100

1463521

21

27

−−

−

210000

60100

1463521

27


Elimination Methods

Step 6 : add suitable multiplies of each row to the rows above to introduce zeros above the leading 1’s.

r2 = 7/2 r3 + r2

r1 = -6r3 + r1

r1 = 5r2 + r1

−

210000

100100

1463521

−

210000

100100

203521

210000

100100

703021


Elimination Methods

1-5 steps produce a row-echelon form (Gaussian Elimination). Step 6 is producing a reduced row-echelon (Gauss-Jordan Elimination).

Remark: Every matrix has a unique reduced row-echelon form, no matter how the row operations are varied. Row-echelon form of matrix is not unique: different sequences of row operations can produce different row- echelon forms.


Back-substitution

Bring the augmented matrix into row-echelon form only and then solve the corresponding system of equations by back-substitution.

Example: [Solved by back substitution]

−

0000000

100000

1302100

0020231

31

31

6

643

5321

132

0223

==++=+−+

x

xxx

xxxx

31

6

643

5321

321

223 1. Step

=−−=

−+−=

x

xxx

xxxx


Back-Substitution

31

6

43

5421

43

31

6

43

5321

31

6

2

243

2 ngSubstituti

2

223

ngSubstituti

2. Step

=−=

−−−=−=

=−=

−+−==

x

xx

xxxx

xx

x

xx

xxxx

x

Step 3. Assign arbitrary values to the free variables [parameters], if any

31

6

5

4

3

2

1

2

243

===

−==

−−−=

x

tx

sx

sx

rx

tsrx


Homogeneous Linear Systems

A system of linear equations is said to be homogeneous if the constant terms are all zero.

Every homogeneous sytem of linear equations is consistent, since all such systems have x1=0,x2=0,...,xn=0 as a solution [trivial solution]. Other solutions are called nontrivial solutions.

01

0

0

221

2222121

1212111

=+++

=+++=+++

nmnmm

nn

nn

xaxaxa

xaxaxa

xaxaxa



Example: [Gauss-Jordan Elimination]

0

02

032

022

543

5321

54321

5321

=++=−−+=+−+−−=+−+

xxx

xxxx

xxxxx

xxxx

⇒

−−−−−

−

000000

001000

010100

010011

011100

010211

013211

010122



The corresponding system of equations is

Solving for the leading variables yields

The general solution is

The trivial solution is obtained when s=t=0

04

53

521

=−=

−−=

x

xx

xxx0

0

0

4

53

521

==+=++

x

xx

xxx

txxtxsxtsx ==−==−−= 54321 ,0,,,



Theorem:

A homogeneous system of linear equations with more unknowns than equations has infinitely many solutions.

1.3 Matrices and Matrix Operations


Matrix Notation and Terminology

A matrix is a rectangular array of numbers with rows and columns.

The numbers in the array are called the entries in the matrix.

Examples:

The size of a matrix is described in terms of the number of rows and columns its contains.

A matrix with only one column is called a column matrix or a column vector.

A matrix with only one row is called a row matrix or a row vector.

[ ] [ ]4,3

1,

000

10

2

,3012,

41

03

21

21

−−

−

πe


Matrix Notation and Terminology

aij = (A)ij = the entry in row i and column j of a matrix A.

1 x n row matrix a = [a1 a2 ... an]

m x 1 column matrix

A matrix A with n rows and n columns is called a square matrix of order n. Main diagonal of A = {a11, a22, ..., ann}

=

nb

b

b

b2

1

nnnn

n

n

aaa

aaa

aaa

21

22221

11211


Operations on Matrices

Definition: Two matrices are defined to be equal if they have the

same size and their corresponding entries are equal. If A = [aij] and B = [bij] have the same size, then A=B if

and only if (A)ij=(B)ij, or equivalently aij=bij for all i and j.

Definition: If A and B are matrices of the same size, then the sum

A+B is the matrix obtained by adding the entries of B to the corresponding entries of A, and the difference A–B is the matrix obtained by subtracting the entries of B from the corresponding entries of A. Matrices of different sizes cannot be added or subtracted.



If A = [aij] and B = [bij] have the same size, then

(A+B)ij = (A)ij + (B)ij = aij + bij and

(A-B)ij = (A)ij – (B)ij = aij - bij

Definition: If A is any matrix and c is any scalar, then the product

cA is the matrix obtained by multiplying each entry of the matrix A by c. The matrix cA is said to be a scalar multiple of A.

If A = [aij], then (cA)ij = c(A)ij = caij.



Definition

If A is an mxr matrix and B is an rxn matrix, then the product AB is the mxn matrix whose entries are determined as follows. To find the entry in row i and column j of AB, single out row i from the matrix A and column j from the matrix B. Multiply the corresponding entries from the row and column together and then add up the resulting products.


Partitioned Matrices


Matrix Multiplication by Columns and by Rows


Matrix Products as Linear Combinations


Matrix Form of a Linear System


Transpose of a Matrix

1.4 Inverses; Rules of Matrix Arithmetic


Properties of Matrix Operations

ab = ba for real numbers a & b, but AB ≠ BA even if both AB & BA are defined and have the same size.

Example:

−

=

−−=

=

−=

03

63,

411

21

03

21,

32

01

BAAB

BA



Theorem: Properties ofa) A+B = B+A

b) A+(B+C) = (A+B)+C

c) A(BC) = (AB)C

d) A(B+C) = AB+AC

e) (B+C)A = BA+CA

f) A(B-C) = AB-AC

g) (B-C)A = BA-CA

h) a(B+C) = aB+aC

i) a(B-C) = aB-aC

Math Arithmetic(Commutative law for addition)

(Associative law for addition)

(Associative for multiplication)

(Left distributive law)

(Right distributive law)j) (a+b)C = aC+bCk) (a-b)C = aC-bCl) a(bC) = (ab)Cm) a(BC) = (aB)C



Proof (d): Proof for both have the same size:

Let size A be r x m matrix, B & C be m x n (same size).

This makes A(B+C) an r x n matrix, follows that AB+AC is also an r x n matrix.

Proof that corresponding entries are equal: Let A=[aij], B=[bij], C=[cij]

Need to show that [A(B+C)]ij = [AB+AC]ij for all values of i and j.

Use the definitions of matrix addition and matrix multiplication.



Remark: In general, given any sum or any product of matrices, pairs of parentheses can be inserted or deleted anywhere within the expression without affecting the end result.

ij

ijij

mjimjijimjimjiji

mjmjimjjijjiij

ACAB

acAB

cacacabababa

cbacbacbaCBA

][

][][

)()(

)()()()]([

22112211

222111

+=+=

+++++++=++++++=+


Zero Matrices

A matrix, all of whose entries are zero, such as

A zero matrix will be denoted by 0 or 0mxn for the mxn zero matrix. 0 for zero matrix with one column.

Properties of zero matrices: A + 0 = 0 + A = A A – A = 0 0 – A = -A A0 = 0; 0A = 0

[ ]0,

0

0

0

0

,0000

0000,

000

000

000

,00

00


Identity Matrices

Square matrices with 1’s on the main diagonal and 0’s off the main diagonal, such as

Notation: In = n x n identity matrix.

If A = m x n matrix, then: AIn = A and InA = A

1000

0100

0010

0001

,

100

010

001

,10

01


Identity Matrices

Example:

Theorem: If R is the reduced row-echelon form of an n x n matrix A, then either R has a row of zeros or R is the identity matrix In.

=

232221

131211

aaa

aaaA

Aaaa

aaa

aaa

aaaAI =

=

=

232221

131211

232221

1312112

10

01

Aaaa

aaa

aaa

aaaAI =

=

=

232221

131211

232221

1312113

100

010

001


Identity Matrices

Definition: If A & B is a square matrix and same size Э AB = BA = I, then A is said to be invertible and B is called an inverse of A. If no such matrix B can be found, then A is said to be singular.

Example:

−

−=

=

31

52,

21

53AB

IBA

IAB

=

=

−

−

=

=

=

−

−=

10

01

31

52

21

53

10

01

21

53

31

52


Properties of Inverses

Theorem: If B and C are both inverses of the matrix A, then B = C.

If A is invertible, then its inverse will be denoted by the symbol A-1.

The matrix

is invertible if ad-bc ≠ 0, in which case the inverse is given by the formula

=

dc

baA

−−−

−−

−=

−

−−

=−

bcad

a

bcad

cbcad

b

bcad

d

ac

bd

bcadA

11


Properties of Inverses

Theorem: If A and B are invertible matrices of the same size, then AB is invertible and (AB)-1 = B-1A-1.

A product of any number of invertible matrices is invertible, and the inverse of the product is the product of the inverses in the reverse order. Example:

=

=

=

89

67,

22

23,

31

21ABBA

−

−=

−

−=

−

−= −−−

2

7

2

934

)(,1

11,

11

23 1

23

11 ABBA

−

−=

−

−

−

−=−−

2

7

2

934

11

23

1

11

23

11AB


Powers of a Matrix

If A is a square matrix, then we define the nonnegative integer powers of A to be

A0=I An = AA...A (n>0) n factors

Moreover, if A is invertible, then we define the negative integer prowers to be A-n = (A-1)n = A-1A-1...A-1

n factors Theorem: Laws of Exponents

If A is a square matrix, and r and s are integers, then ArAs = Ar+s = Ars

If A is an invertible matrix, then A-1 is invertible and (A-1)-1 = A An is invertible and (An)-1 = (A-1)n for n = 0, 1, 2, ... For any nonzero scalar k, the matrix kA is invertible and

(kA)-1 = 1/k A-1.


Powers of a Matrix

Example:

=

31

21A

−

−=−

11

231A

=

=

4115

3011

31

21

31

21

31

213A

−

−=

−

−

−

−

−

−== −−

1115

3041

11

23

11

23

11

23)( 313 AA


Polynomial Expressions Involving Matrices If A is a square matrix, m x m, and if

is any polynomial, then we define

Example:

nnxaxaaxp +++= 10)(

nnAaAaIaAp +++= 10)(

432)( 2 +−= xxxp

=

+

−−

=

+

−−

−=+−=

130

29

40

04

90

63

180

82

10

014

30

213

30

212432)(

2

2 IAAAp


Properties of the Transpose

Theorem: If the sizes of the matrices are such that the stated operations can be performed, then

a) ((A)T)T = Ab) (A+B)T = AT + BT and (A-B)T = AT – BT

c) (kA)T = kAT, where k is any scalard) (AB)T = BTAT

The transpose of a product of any number of matrices is equal to the product of their transpose in the reverse order.


Invertibility of a Transpose

Theorem: If A is an invertible matrix, then AT is also invertible and (AT)-1 = (A-1)T

Example:

−−

=

−−

=

−−

=

−−

=

−−=

−−−

53

21)(,

53

21)(,

52

31

13

25,

12

35

111 TT

T

AAA

AA


Exercise

Show that if a square matrix A satisfies A2-3A+I=0, then A-1=3I-A

Let A be the matrix

Determine whether A is invertible, and if so, find its inverse. [Hint. Solve AX = I by equating corresponding entries on the two sides.]

110

011

101

1.5 Elementary Matrices and

a Method for Finding A-1


Elementary Matrices

Definition: An n x n matrix is called an elementary matrix if

it can be obtained from the n x n identity matrix In by performing a single elementary row operation.

Example:

1. Multiply the second row of I2 by -3.

2. Interchange the second and fourth rows of I4.

3. Add 3 times the third row of I3 to the first row.

−

100

010

301

:3,

0010

0100

1000

0001

:2,30

01:1


Elementary Matrices

Theorem: (Row Operations by Matrix Multiplication) If the elementary matrix E results from performing a certain

row operation on Im and if A is an m x n matrix, then the product of EA is the matrix that results when this same row operation is performed on A.

Example:

EA is precisely the same matrix that results when we add 3 times the first row of A to the third row.

−=

=

−=

01044

6312

3201

103

010

001

,

0441

6312

3201

EA

EA


Elementary Matrices

If an elementary row operation is applied to an identity matrix I to produce an elementary matrix E, then there is a second row operation that, when applied to E, produces I back again.

Inverse operation

Row operation on I that produces E

Row operation on E that reproduces I

Multiply row i by c ≠ 0 Multiply row i by 1/c

Interchange rows i and j Interchange rows i and j

Add c times row i to row j Add –c times row i to row j


Elementary Matrices

Theorem: Every elementary matrix is invertible, and the inverse is also an elementary matrix.

Theorem: (Equivalent Statements) If A is an n x n matrix, then the following

statements are equivalent, that is, all true or all false.a) A is invertible

b) Ax = 0 has only the trivial solution.

c) The reduced row-echelon form of A is In.

d) A is expressible as a product of elementary matrices.


Elementary Matrices

Proof:

Assume A is invertible and let x0 be any solution of Ax=0.

Let Ax=0 be the matrix form of the system

)()()()()( adcba ⇒⇒⇒⇒)()( ba ⇒

0,0,0)(,0)( 00011

01 ==== −−− xIxxAAAAxA

)()( cb ⇒

0

0

11

1111

=++

=++

nnnn

nn

xaxa

xaxa

⇒

010000

00100

00010

00001

0

0

0

21

22221

11211

nnnn

n

n

aaa

aaa

aaa


Elementary Matrices

Assumed that the reduced row-echelon form of A is In by a finite sequence of elementary row operations, such that:

By theorem, E1,…,En are invertible. Multiplying both sides of equation on the left we obtain:

This equation expresses A as a product of elementary matrices.

If A is a product of elementary matrices, then the matrix A is a product of invertible matrices, and hence is invertible.

Matrices that can be obtained from one another by a finite sequence of elementary row operations are said to be row equivalent.

An n x n matrix A is invertible if and only if it is row equivalent to the n x n identity matrix.

)()( dc ⇒

nk IAEEE =12

112

11

112

11

−−−−−− == knk EEEIEEEA

)()( ad ⇒


A Method for Inverting Matrices

To find the inverse of an invertible matrix, we must find a sequence of elementary row operations that reduces A to the identity and then perform this same sequence of operations on In to obtain A-1.

Example:

Adjoin the identity matrix to the right side of A, thereby producing a matrix of the form [A|I]

Apply row operations to this matrix until the left side is reduced to I, so the final matrix will have the form [I|A-1].

=

801

352

321

A



Added –2 times the first row to the second and –1 times the first row to the third.

Added 2 times the second row to the third.

Multiplied the third row by –1.

Added 3 times the third row to the second and –3 times the third row to the first.

We added –2 times the second row to the first.

−−−−

−

−−−−

−

−−−

−

−−

−−

−−

−−

125

3513

91640

100

010

001

125

3513

3614

100

010

021

125

012

001

100

310

321

125

012

001

100

310

321

101

012

001

520

310

321

100

010

001

801

352

321



Often it will not be known in advance whether a given matrix is invertible.

If elementary row operations are attempted on a matrix that is not invertible, then at some point in the computations a row of zeros will occur on the left side.

Example:

Added -2 times the first row to the second and

added the first row to the third.

Added the second row to the third.

−−=

521

142

461

A

−−−−

−−−

−−

111

012

001

000

980

461

101

012

001

980

980

461

100

010

001

521

142

461


Exercises

Consider the matrices

Find elementary matrices, E1, E2, E3, and E4, such that

a. E1A=B

b. E2B=A

c. E3A=C

d. E4C=A

−−−=

−−=

−−=

372

172

143

,

143

172

518

,

518

172

143

CBA


Exercises

Express the matrix:

in the form A = E F G R, where E, F, G are elementary matrices, and R is in row-echelon form.

−−− 8152

8331

8710

1.6 Further Results on

Systems of Equations and Invertibility


Linear Systems

Theorem:

Solving Linear Systems by Matrix Inversion:

If A is an invertible n x n matrix, then for each n x 1 matrix b, the system of equations Ax = b has exactly one solution, namely, x = A-1b.

Linear systems with a common coefficient matrix.Ax=b1, Ax=b2, Ax=b3, ..., Ax=bk

If A is invertible, then the solutions

x1=A-1b1, x2=A-1b2, x3=A-1b3, ..., xk=A-1bk

This can be efficiently done using Gauss-Jordan Elimination on [A|b1|b2|...|bk]


Linear Systems

Example: (a) (b)

The solution:

(a) x1=1, x2=0, x3=1

(b) x1=2, x2=1, x3=-1

98

5352

432

31

321

321

=+=++=++

xx

xxx

xxx

68

6352

132

31

321

321

−=+=++=++

xx

xxx

xxx

−

−

1

1

2

1

0

1

100

010

001

6

6

1

9

5

4

801

352

321


Properties of Invertible Matrices

Theorem: Let A be a square matrix.a) If B is a square matrix satisfying BA=I, then B=A-1.

b) If B is a square matrix satisfying AB=I, then B=A-1.

Theorem: Equivalent Statementsa) A is invertible

b) Ax=0 has only the trivial solutions

c) The reduced row-echelon form of A is In

d) A is expresssible as a product of elementary matrices

e) Ax=b is consistent for every n x 1 matrix b

f) Ax=b has exactly one solution for every n x 1 matrix b


Properties of Invertible Matrices

Theorem: Let A and B be square matrices of the same size. If AB is invertible, then A and B must also be invertible.

A fundamental problem.

Let A be a fixed m x n matrix. Find all m x 1 matrices b such that the system of equations Ax=b is consistent.


Exercises

Solve the system by inverting the coefficient matrix.

Find condition that b’s must satisfy for the system to be consistent.

6642

4973

744

032

=−−−−=+++=+++=−−−

zyxw

zyxw

zyxw

zyx

221

121

23

46

bxx

bxx

=−=−

1.7 Diagonal, Triangular,

and Symmetric Matrices


Diagonal Matrices

A square matrix in which all the entries off the main diagonal are zero. Example:

A diagonal matrix is invertible if and only if all of its diagonal entries are nonzero.

−

−

8000

0000

0040

0006

,

100

010

001

,50

02

=

nd

d

d

D

00

00

00

2

1

=−

nd

d

d

D

100

010

001

2

1

1

=

kn

k

k

k

d

d

d

D

00

00

00

2

1


Diagonal Matrices

Example:

−=

200

030

001

A

−=−

2

100

03

10

0011A

−=

200

030

0015A

−=−

321

24315

00

00

001

A

=

=

333322311

233222211

133122111

3

2

1

333231

232221

131211

343333323313

242232222212

141131121111

34333231

24232221

14131211

3

2

1

00

00

00

00

00

00

adadad

adadad

adadad

d

d

d

aaa

aaa

aaa

adadadad

adadadad

adadadad

aaaa

aaaa

aaaa

d

d

d


Triangular Matrices

Lower triangular = a square matrix in which all the entries above the main diagonal are zero.

Upper triangular = a square matrix in which all the entries under the main diagonal are zero.

Triangular = a matrix that is either upper triangular or lower triangular.

44

3433

242322

14131211

000

00

0

a

aa

aaa

aaaa

44434241

333231

2221

11

0

00

000

aaaa

aaa

aa

a


Triangular Matrices

Theorem: (basic properties of triangular matrices) The transpose of a lower triangular matrix is upper

triangular, and the transpose of an upper triangular matrix is lower triangular.

The product of lower triangular matrices is lower triangular, and the product of upper triangular matrices is upper triangular.

A triangular matrix is invertible if and only its diagonal entries are all nonzero.

The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular.


Triangular Matrices

Example:

The matrix A is invertible, since its diagonal entries are nonzero, but the matrix B is not.

This inverse is upper triangular. This product is upper triangular.

−

−=

−=

100

100

223

,

500

420

131

BA

−−=

−

−=−

500

200

223

,

00

0

1

51

52

21

57

23

1 ABA


Symmetric Matrices

A square matrix A is called symmetric if A = AT.

A matrix A = [aij] is symmetric if and only if aij=aji for all values of i and j.

−

−

−

4

3

2

1

000

000

000

000

,

705

034

541

,53

37

d

d

d

d

−

705

034

541


Symmetric Matrices

Theorem: If A and B are symmetric matrices with the same size, and if k is any scalar, then AT is symmetric A+B and A-B are symmetric kA is symmetric

Theorem: If A is an invertible matrix, then A-1 is

symmetric. If A is an invertible matrix, then AAT and ATA

are also invertible.


Exercise

Find all values of a, b, and c for which A is symmetric.

Find all values of a and b for which A and B are both not invertible.

−+

+++−=

720

53

2222

ca

cbacba

A

−−

=

−+=

7320

05,

30

01

baB

baA

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